As shown in [1]} such specific weights are crucial to obtain the lower semicontinuity of the energy \(\mathcal {F}\)  in the subfamily \(\mathcal {C}_m\)  of configurations with free-crystal regions presenting a fixed number \(m\in \mathbb {N}\)  of boundary connected components, namely
\(\mathcal {C}_m:=\Big \lbrace (A,u)\in \mathcal {C}:\,\,\text{$\partial A$ has at most $m$ connected components} \Big \rbrace ,\) 
